Many people have concluded that the balanced outputs are superior to the unbalanced outputs.
For those of us with unbalanced amplifiers, it seems that a XLR balanced to RCA unbalanced cable would extract much of the benefit. I see that the balanced circuit uses one op amp for each left and right channel, while the unbalanced circuit uses a single dual op amp. This could explain part of the improvement.

The balanced output is superior to the unbalanced only when one use that with an unbalanced input amplifier. There is no any benefit to convert that balanced to unbalanced for use it so with a conventional amplifier. Using RCA output with an unbalanced amplifier is the only reasonable way and it have the max benefit for this type of output.
The logic about how many opamp are used to process the signal on one or another type of output, is not quite right. The right logic it may be: the less components on the signal path, the better for the integrity/quality of that signal...
Converting the balanced to unbalanced it cancel all the benefit of the balanced output. If one have only the balanced output on a device and it dispose only an unbalanced amplifier, then a such convertor do the job to connect those together and use it the system. But this is a temporary compromise solution, not something which it "would extract much of the benefit" of an balanced output...
My suggestion is to use it the right way and be happy with that benefit you can get so...

Is right like this: The balanced output is superior to the unbalanced only when one use that with an balanced input amplifier.

I think is not quite well that this forum do not permit posts editing, but a half hour after the post is first published. Having continuous edit access to his posts, one may correct many of the published posts or just delete what it become unnecessary, and so it will be prevented too much (wrong) text in the threads...

Yes corus, I think you are right. On reflection last night I realized that one would probably need a transformer to convert the signal, as the two halves should be combined (The signal is probably unbalanced straight from the DACs). (Although as far as pickup,I should note that my cable is only 3 feet long). So how about if I do this using a transformer on the amplifier side, or should I instead think about ways to improve the unbalanced output?

Actually there are many other improvements one may have in place before thinking to convert balanced to unbalanced...
I too, do not have yet an balanced amplifier, and I have used all the time the unbalanced one. I have to say that I still have a very good sound this way. Anyway I want to switch to balanced amplifier. Is my priority now. I think is the right way I may use all the benefits of such output/connection.
So, my suggestion is you try (to forget that way with an transformer and so...) to get the most out of RCA output, doing the rest of quite many other mods which it may bring very real and important improvements in that unbalanced way. BTW, the most of the mods one may want to put in place, have common improvements for both ways: balanced/unbalanced. Then think to get an balanced amplifier... to go balanced with an already good modded device.

I'm using the BDP-105 with a transformer volume control (TVC), which can take balanced as well as single ended inputs with precisely the same hardware. I also think the balanced outputs on the Oppo are slightly better, and a TVC preamp is probably one of the best and most cost effective ways to utilise it. You can then still use a single ended power amp. Additional advantage is that you can break any ground loops, and TVC's can sound extremely good.